{"paper":{"title":"Boundary regularity for the second boundary-value problem of Monge-Amp\\`ere equations in dimension two","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Jiakun Liu, Shibing Chen, Xu-Jia Wang","submitted_at":"2018-06-25T14:23:03Z","abstract_excerpt":"In this paper, we introduce an iteration argument to prove that a convex solution to the Monge-Amp\\`ere equation $\\mbox{det } D^2 u =f $ in dimension two subject to the natural boundary condition $Du(\\Omega) = \\Omega^*$ is $C^{2,\\alpha}$ smooth up to the boundary. We establish the estimate under the sharp conditions that the inhomogeneous term $f\\in C^{\\alpha}$ and the domains are convex and $C^{1,\\alpha}$ smooth. When $f\\in C^0$ (resp. $1/C<f<C$ for some positive constant $C$), we also obtain the global $W^{2,p}$ (resp. $W^{2,1+\\epsilon}$) regularity."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1806.09482","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}